Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^2}$ is a real number and $|\alpha - \beta| = 2 \sqrt{3}.$  Find $|\alpha|.$
Let $\alpha = x + yi$ and $\beta = x - yi.$  Then from $|\alpha - \beta| = 2 \sqrt{3},$ $2|y| = 2 \sqrt{3},$ so $|y| = \sqrt{3}.$

Now, $\frac{\alpha}{\beta^2}$ is real.  Since $\alpha$ and $\beta$ are complex conjugates, $\alpha^2 \beta^2$ is real, so $\frac{\alpha}{\beta^2} \cdot \alpha^2 \beta^2 = \alpha^3$ is real.  But
\[\alpha^3 = (x + yi)^3 = (x^3 - 3xy^2) + (3x^2 y - y^3) i,\]so $3x^2 y - y^3 = 0,$ or $y(3x^2 - y^2) = 0.$  Since $|y| = \sqrt{3},$ $y \neq 0,$ so $3x^2 = y^2 = 3,$ which means $x^2 = 1.$  Therefore,
\[|\alpha| = \sqrt{x^2 + y^2} = \sqrt{1 + 3} = \boxed{2}.\]